Towards a Grand Unified Theory of Mathematics and Physics

A draft of an essay I’ve written, with plans to submit it to the FQXI essay contest, is available here. Constructive comments welcome…

People who have a take on the subject that has nothing to do with what I’m writing about are encouraged to submit their own essays to FQXI, but not to post them here.
Update: Thanks to all commenters for often helpful comments. I’ve revised the essay a bit, mostly by adding some material at the end, material that to some extent addresses important issues raised by some commenters.
Update: The essay has been submitted and is posted here.

64 Responses to Towards a Grand Unified Theory of Mathematics and Physics

“Finally, it would seem to be that mathematics is completely abstract but limited (by our own intellects) whereas the physical world is somewhat less abstract and likely somewhat less limited.”

I would say it is the other way around. In Mathematics, you 1) Select objects that allow some form of arrangement 2) Posit invariants over these arrangements 3) Select a Logic (itself a subject and part of of Mathematics) 4) See where the Logic leads you, trying to steer clear of breakdowns (logical contradictions)

Given that humans explore the above structure, exploration is necessarily limited to symbolic computation by Turing machines. So far, this has not been an area of concern or we can work around the hang-ups nicely (Uncomputable number found? Just call it \Omega and press on)

The physical world seems to implement a particular structure that is amenable to explorations of the above kind. The universe can be described by a finite substructure of itself, which is nice! Why is this so and whether this description has some remarkable property (e.g. it may an “island in the space of consistent theories”, not need any “arbitrary values” or may be the only use in need of some “highest order structure”) are of course open questions.

I sure hope it has remarkable properties. Getting some arbitrary ooze out of all the work would be a let-down.

Most mathematicians are of the opinion that deep down at the fundamental level of nature there is just numbers nothing else but numbers.This is totally mistaken since mathematics by definition is an empirical science. It born out of observations and experiments on reality. Differential geometry and its other forms are born from the properties of space-time. By making more profound studies of the physical properties of space-time new mathematics arises.This approach led Newton to discover calculus. If one on the other hand tries to start with abstract mathematics and hopes to explain reality one ends up with String Theory. A theory with no predictive power nor a connection with reality.

Stuart, can you cite a modern and respectable mathematician who expressed the view that “deep down at the fundamental level of nature there is just numbers nothing else but numbers”? Numbers are only elements of a very particular mathematical structure, that of a field, and there are much more algebraic structures which are richer our even prettier than “numbers”. Why should nature be in a fundamental level a particular mathematical structure, be it a field or anything else? Also, you can construct alternative mathematics using alternative logic (fuzzy or quantum logics for example). What kind of mathematics is nature made off if any at all?

Now, the view that “mathematics by definition is an empirical science. It born out of observations and experiments on reality” is a very naive and simplistic one of mathematics. It may be true that historically mathematics grew out from empirical observations, but mathematical truth is completely independent from physical experiments, and you do not even need to be a platonist to see this. I do not think that Grothendieck ever needed directly an experimental result to produce any discovery in algebraic geometry.

This idea of a unified theory of mathematics and physics seem very presumptuous to me, to say the least. Even if a better understanding of the mathematical structure of the Standard Model, as discussed by Peter in his text, or of a theory of quantum gravity based on string theory, as defended by Michael Rio in his own essay (1502.04794v1), can produce new non-trivial applications of sophisticated pieces of mathematics, motivate new conjectures, new insights to the Langlands program &c., there will still exists many other mathematical theories outside such ‘unification’ that, nevertheless, mathematicians still regards as interesting. So is this a really unification of mathematics?

For example, the Greeks could very well consider the theory of statics of Archimedes as an unification of physics and mathematics (indeed, they not even separated physics, mathematics and philosophy at all), since it is an application of Euclidean geometry, which is basically the whole of Greek mathematics, to describe physical configurations.

I think that the most interesting part of Peter’s essay is the relation of number theory and QFT, but which could very well be much more developed. Well, maybe in another opportunity!

Stuart,
“If one on the other hand tries to start with abstract mathematics and hopes to explain reality one ends up with String Theory.”
You really can’t blame string theory on the mathematicians. It was invented by physicists in the late sixties who had little interest in mathematics, pursued by physicists during the seventies and early eighties who had little interest in mathematics, became popular when someone influential decided it was a promising physical idea about unification (yes, he happens to also have done work with great mathematical significance), and string theory unification is now based on a physical idea (the landscape) which is completely mathematically empty.

Erik,

I saw that. It’s a good example of what I don’t have in mind: the idea that some piece of abstract mathematics applied to a conjectural framework with no known connection to fundamental physics is the way forward. I try to make clear that what I have in mind is mathematics applied to a deeper understanding of the specific QFT that has been successful. I don’t think the idea that applying some abstract math will tell you what M-theory is and show how to do unification is at all a promising one. Lots of effort has gone in that direction in the last twenty years, with nothing to show for it.

Radioactive,
Actually, as far as telling us anything about fundamental physics (as opposed to dust), BICEP2 IS just hype. Despite the endless claims of the last ten years about how Planck data would tell us about fundamental physics, I’m having trouble seeing how it has changed anything in our understanding of physical laws. About LHC2, one can hope for the best, but despite the hype, I think the most physicists feel that the odds now are that it will not see anything unexpected.

To put the reason for interest in number theory in a language that physicists might find more palatable, the claim is just that progress in fundamental physics has often come in the past from a better understnding of symmetries and how to exploit them. Mathematicians know a lot about symmetries that has not yet found application in physics (representation theory is a large field), and the Langlands program is currently one of the most active areas in mathematics where new work is being done on the exploitation of symmetries.

If you haven’t looked at the latest version of the essay, see material I’ve added at the beginning of the concluding section which addresses the “unification of mathematics” question. But yes, the essay is intended to be somewhat presumptuous and provocative.

I do hope to someday get something serious written about qft and number theory, but that’s a big project, and I think our understanding is still very fragmentary.

“The universe can be described by a finite substructure of itself, which is nice!”

I don’t think that’s been determined yet, without the word ‘partially’ in there before ‘described’. Can it be, though? I think that it’s a very interesting question. I personally think the answer is ‘No, it can’t be’. I suppose if physical reality is actually infinite in extent, then it sort of follows that it could contain another arbitrarily large subset, but I still wonder. What does it mean to describe something completely? Don’t you essentially have to make an actual copy, and thus break the no-cloning quantum theory constraint?

It is interesting to note, though, that what we think of as mathematics is being performed by a bunch of ionic charges skittering around in a lump of water and protein. If math really is ‘larger’ than physical reality, which I gather most of you think, it makes me doubtful that we would ever be able to apprehend it fully with our lumps of protein.

I suppose you and I have different outlooks. I am still willing to wait for the results of experiments, overhyped or not, null or not, because they provide grist to the mill of high quality theoretical physics. Whatever the right theory of fundamental physics at high energy scales, there is no guarantee that it will be mathematically elegant but it certainly will predict the results of experiments.

Speaking of things that have been overhyped, I have heard uncountable times how advanced representation theory/topology/number theory/something else under the mathematical sun is going to lead to a new understanding of fundamental physics. This essay is speculation that the Langlands program is going to help fundamental physics because it will help us unify everything back to the one holy mathematical Object. It sounds like ancient philosophy instead of natural science. The “unreasonable effectiveness of mathematics” itself is overhyped and you are helping to build it.

If theorists want to sit on their hands and wait for new experimental results that may be a long time coming (as in, after they’re dead…) they can do that. One problem with this tactic is that their colleagues may decide to stop hiring more theorists, and there is some data indicating that is happening, see

If there isn’t any good reason to hire theorists then why hire them? It isn’t a choice between becoming a hep-th postdoc and working in a coal mine. Perhaps fewer garbage papers on the arxiv per day would be good for the field.

Since Anglo-Saxon studies and Abstract Topology departements continue to exist in universities I doubt we’ll ever see a complete extinction of mathemathical physicists, though there may be fewer of them, which is not a great loss.

“Speaking of things that have been overhyped, I have heard uncountable times how advanced representation theory/topology/number theory/something else under the mathematical sun is going to lead to a new understanding of fundamental physics.”

Yes, Herr Stark, I agree! The work of Hermann Weyl belongs more to representation theory, topology, and number theory than physics. I mean, really- a gauge theory? Pauli has already shown these structures to be as unphysical as Koenig’s spin structure idea.

We should focus on physics of immediate impact in defense and industry- Indeed, Herr Stark, any new physics will manifest through your famous shifts in ultracold atoms. As to whatever Bardeen, Einstein, Oppenheimer, Noether and Weyl are erestwhile up to right now, I’m sure none of it will ever pan out. I mean, really, Herr Stark, who cares about understanding the fractional quantum hall effect? Topological order, edge effects?

What we need is still more experiment. Physics should just be all experimentalists, really- no need to even come up with models to verify! Who needs models, let’s just run experiments! Then, something actually interesting comes up in those experiments, we can start developing the mathematical physics required to understand the new physical phenomena. We will simply call some of the mathematical physicists back from the coal mines.

It is probably human nature to think one’s own post had something to do with your changes, I have no idea if that’s the case, but I like that you’d subtly distanced yourself from the String Theory antics of the past 30+ years. That’s probably a good thing.

Your point about Einstein’s insight on GR being helped by Riemannian
geometry was interesting as well.

I sort of wish I could see the old essay, but that might be a chore. I will suffer my personal hallucinations as to what was actually changed…

Nice essay! I also think there are interesting things to discover by combining number theory and QFT. The connection between the function field (over C) case of the Langlands program and QFT, is by now well established. However for number fields the situation seems rather unclear. There are some intriguing observations though regarding this hypothetical NT/QFT connection. For instance the Weil distribution whose positivity implies the Riemann Hypothesis looks a lot like the (multiplicatively translation-invariant) two point function of a Euclidean QFT. This was pointed out in a paper by Burnol: http://arxiv.org/abs/math/9809119
Also, Connes’ trace formula in http://arxiv.org/abs/math/9811068
involves taking a limit of removing both UV and IR cut-offs just as in the usual approach for constructing a QFT model. Clearly the Weil distribution has to do with
the action of rescaling group R^*, but the question is: on what?
A possible connection to such an action on QFTs, namely the renormalization group, was considered in this article by Leichtnam: http://arxiv.org/abs/math/0603576
Having studied Euclidean QFT over both R which is the usual setting, and over Q_p, I can say that the definitions and basic properties are very natural in both of these settings and they parallel each other in a very nice way. For instance, a good theory of probability measures on spaces of distributions require a nuclear topological vector space. Over R it is S(R^d) which is countably Hilbert. Over it is S(Q_p^d)
which is not metrizable. These two spaces are in some sense at the opposite ends of the spectrum ofthe general notion of nuclear space due to Grothendieck. All the good theorems like Bochner-Minlos (for constructing free boson models) or the Levy continuity, hold for these two extreme examples. In fact the theory over p-adics
is simpler than over R and therefore offers a useful testing ground of ideas for constructing QFTs rigorously. I explained that in my short “essay” http://arxiv.org/abs/1311.4897